Can You Solve This Complex Differential Equation?

[Ben-CS]

I have decided to post an exercise and see what happens. You can post your own problems, too, if you wish. Have fun with this one!


Solve the following differential equation:

y′′′ = y

 

[arcnets]

y(t) = a*exp(b*t) where b^3 = 1.

 

[dg]

The general solution is of the form

y(x)=c1*exp(q1*x)+c2*exp(q2*x)+c3*exp(q3*x)

where c1, c2 and c3 are (complex) constant that can be fixed by initial conditions and
q1=1
q2=(-1+i*sqrt(3))/2
q3=(-1-i*sqrt(3))/2

This can be re-written in the real field as

y(x)=exp(x)* c1 + A*exp(-x/2)*cos(sqrt(3)*x+B) )

where A and B are new constants depending on c2 and c3 only.